While the Shannon-Stam entropy power inequality is a powerful tool in the study of convolutions of probability densities on \mathbb{R}^n, the search for a satisfactory discrete analogue of the same that is sharp has long been largely fruitless, with only some rather specialized results available. We prove several simple and sharp new lower bounds for the Rényi entropies of the convolution of probability distributions on the integers in terms of certain (discrete) rearrangements of these distributions. These inequalities may be thought of as discrete entropy power inequalities for integer-valued random variables. Furthermore, they provide a unification of the Cauchy-Davenport theorem on the integers from additive combinatorics, as well as of an influential lemma due to Littlewood-Offord and Erdos (which it significantly generalizes). If time permits, we will discuss extensions to cyclic groups of prime order. The talk is based on joint work with Liyao Wang (J. P. Morgan) and Jae Oh Woo (University of Texas, Austin).

In this thesis, I study a variety of inf-convolution operators and their applications to a class of general transportation inequalities, more specifically in the graphs. We prove that some inf-convolution operators are solutions of a Hamilton-Jacobi’s inequation. We deduce from this result some properties concerning different functional inequalities, including Log-Sobolev inequalities and weak-transport inequalities.